#6687019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,9,11)(7,12,8,10)(13,14,17,20)(15,23,27,19)(16,28,29,21)(18,22,31,34)(24,32,26,36)(25,33,39,37)(30,41,42,35)(38,44,40,43);(5,6)(7,10)(8,12)(9,11)(13,15,17,27)(14,19,20,23)(16,25,29,39)(18,24,31,26)(21,33,28,37)(22,36,34,32)(30,38,42,40)(35,44,41,43);(1,2,3,4)(14,20)(15,24)(18,31)(19,32)(21,28)(23,36)(25,38)(26,27)(30,42)(33,43)(37,44)(39,40);(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,28)(22,35)(23,37)(24,38)(26,40)(27,39)(31,42)(32,43)(34,41)(36,44);(5,8,9,7)(6,10,11,12)(15,26)(16,29)(18,31)(19,32)(21,28)(22,34)(23,36)(24,27)(25,38)(33,44)(37,43)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594215);